Thank you whosnext. Very good effort indeed.

I revisited some of those equations and have doubts about equation for c3a

It doesn't include conditional probability that player 2 folded his hand.

Should it be like this?

C3a * (c3a-c1)/(1-c1) * 1/c2 = 1/3





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Edit : it seems the first set of equations were correct.

P (p3 wins against p2 | p2 folded) = 1

In that case, solutions provided by whosnext are correct and we have the answers for a 4 player game.

The size of the solution for n players ( including the blind player) is 2 ^ (n-2)

I verified all the equations before I did my solution journey. (Not to diminish the derivation, but it is quite straightforward.)

The 4-player solution with antes is yet to be derived, though, as in the 3-player case, the underlying equations are unchanged except for the constant (the pot odds part).

The 5-player no-ante solution is probably straightforward to derive (mix of analytical and numerical).

To be honest, I am not sure what these GTO solutions of this highly stylized toy game reveal to us.

Here is the solution for the 5-player game without antes. Following the lead from above, label the players P0, P1, P2, P3, and P4 (P0 bets blind 100% of the time). The other players can call or fold. The GTO solution consists of the minimum hand each player calls with on each possible branch of the game tree. As above, the GTO solution for many cases in the 5-player game tree devolve into the lower-order GTO solutions for the smaller games (derived and presented previously in the thread).

The "new" GTO branches are as follows:

C1 = .76996520 (P1's min call hand)

C2 = .87267090 (P2's min call hand when P1 calls)

C3A = .86606407 (P3's min call hand when P1 calls and P2 folds)

C3B = .92579529 (P3's min call hand when P1 calls and P2 calls)

C4A = .85920815 (P4's min call hand when P1 calls, P2 folds, P3 folds)

C4B = .92130617 (P4's min call hand when P1 calls, P2 folds, P3 calls)

C4C = .92408574 (P4's min call hand when P1 calls, P2 calls, P3 folds)

C4D = .95543466 (P4's min call hand when P1 calls, P2 calls, P3 calls)

Incredible work whosnext. Thank you.

I tried solving this in python and it was taking forever.

Here are some of my observations

1. The effect of position is less pronounced than I was expecting it to be.

When facing the same shoving range and getting same pot odds, players in later position only defend slightly more often than players in earlier positions ( 0.006 seems to be gap in the 5 player solution)

2. The effect of an extra player putting money in the pot is huge. Even though the next to act players are now getting better odds, they play much tighter than they were playing looser in the above observation.

3. Player 1's open shoving ( in this case calling) range is getting tighter the more players are in the pot. Very expected. But the jumps are smaller for each increment in game size. ( 0.63 for 3 player game, 0.71 for 4 player game and 0.77 for 5 player game) I wonder if there is a natural limit to it. That is no matter how many players are in the pot, player 1 can always shove top 1% of hands or something. Probably not true.

There should be a natural convergence for player 1 given infinite extra participants. Also, if player 1 could calculate this, and demonstrate the strategy to the rest of the world, then if player 1 actually limped then the game would be over, because any limp thereafter would be losing EV versus the first limper. This is what interests me in the infinite extension of these types of toy games. A first move that is “a look to the sky” that counters all future moves in infinite extension.

Cmon. Suppose there were a billion players and every opponent played the top 100 millionth of their hands

Assuming antes, if first limper is optimal, then there will be no EV left for anyone else.

Thinking about infinite number of players may be counter productive to this thread. I apologise if it derailed the thread. I will post some of my final thoughts related to this topic and it may be better to start a new thread for the purely theoretical infinite player game. Please let me know if that's better.

Instead of one player playing blind, let's say you have n players playing blind. ( or 1 player playing blind and the other n-1 players using a 100% shoving range)

Strategy for the last player in n+1 player pool then is (1/1+n) ^(1/n) -> 1 as n goes to inf.

So if you had everyone going all in blind, the last player can't call.

If you flip the script, and say you knew everyone left to act after you is a degenerate gambler and will always call, as the first to act player you can't call anything.

If you knew everyone left to act behind you is always folding, you can call with 0.5

So what should happen when you know everyone left to act behind you is neither a degenerate gambler nor a folding machine but an optimal player.

If the answer is still to fold everything, then we are saying there is no difference between facing infinite optimal opponents and facing infinite degenerate gambling opponents.
( which may be true but is definitely insightful)

Very nice. So what is each of the players EVS before the game starts?

Here is the solution for the 6-player game without antes.

Following the lead from above, label the players P0, P1, P2, P3, P4, and P5 (P0 bets blind 100% of the time). The other players can call or fold. The GTO solution consists of the minimum hand each player calls with on each possible branch of the game tree. As above, the GTO solution for many cases in the 6-player game tree devolve into the lower-order GTO solutions for the smaller games (derived and presented previously in the thread).

The "new" GTO branches are as follows:

---

C1 = .80694545 (P1's min call hand)

---

C2 = .89426354 (P2's min call hand when P1 calls)

---

C3A = .88967823 (P3's min call hand when P1 calls and P2 folds)

C3B = .93873171 (P3's min call hand when P1 calls and P2 calls)

---

C4A = .88493582 (P4's min call hand when P1 calls, P2 folds, P3 folds)

C4B = .93566347 (P4's min call hand when P1 calls, P2 folds, P3 calls)

C4C = .93755983 (P4's min call hand when P1 calls, P2 calls, P3 folds)

C4D = .96332643 (P4's min call hand when P1 calls, P2 calls, P3 calls)

---

C5A = .88006664 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 folds)

C5B = .93240807 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 calls)

C5C = .93439073 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 folds)

C5D = .96139927 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 calls)

C5E = .93637226 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 folds)

C5F = .96250749 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 calls)

C5G = .96296510 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 folds)

C5H = .97752185 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 calls)

How did you calculate theee values? Did you write a program to solve/estimate systems of equations? Did you solve them yourself? Did you use some recursive equilibrium search algorithm?

I do not have any special software. I wrote some pretty basic functions to numerically find the solutions. It helps that some of the equations are solvable analytically (one variable in terms of one or more others). Also, even when equations are not analytically solvable, due to the nature of the game tree the variables "cascade" (sometimes in pairs) to a significant degree. So it is essentially just a matter of intelligently and efficiently searching at the highest levels and letting the possible solutions cascade down until all of them are satisfied.

Here are the EV's for the GTO players in the respective toy games (without antes) solved so far.

Players	P0	P1	P2	P3	P4	P5
2	-0.250000	0.250000
3	-0.399056	0.167931	0.231125
4	-0.496559	0.125292	0.161640	0.209627
5	-0.565348	0.099747	0.122850	0.152260	0.190491
6	-0.616474	0.082576	0.098702	0.118191	0.142604	0.174401

Very nice again. The answers seem logical and refute the illogical notion that the first guy after the blind is in some way not in the second worst spot.

Since you are so good at this here three more questions. (I hope you are simulating rather than calculating.)

With various number of players how often does the best hand win? (Obviously with two players the answer is 7/8.)

For each player, what percentage of the time that it does indeed have the best hand does it still not win?

For each player what percentage of the time does it win with the not best hand?

Hopefully people realize that I simply wanted a simulation for the three questions above using the assumption that whosnext's results in post #36 are correct. Just tell your computer to deal out a billion hands using those rules and see who has the best hand and who actually wins. I don't want to dig out my Timex Sinclair and reread the three pages of instructions on basic Basic.

Here is the solution for the 7-player game without antes.

Following the lead from above, label the players P0, P1, P2, P3, P4, P5, and P6 (P0 bets blind 100% of the time). The other players can call or fold. The GTO solution consists of the minimum hand each player calls with on each possible branch of the game tree. As above, the GTO solution for many cases in the 7-player game tree devolve into the lower-order GTO solutions for the smaller games (derived and presented previously in the thread).

The "new" GTO branches are as follows:

---

C1 = .83386680 (P1's min call hand)

---

C2 = .90968767 (P2's min call hand when P1 calls)

---

C3A = .90632926 (P3's min call hand when P1 calls and P2 folds)

C3B = .94788043 (P3's min call hand when P1 calls and P2 calls)

---

C4A = .90286674 (P4's min call hand when P1 calls, P2 folds, P3 folds)

C4B = .94565715 (P4's min call hand when P1 calls, P2 folds, P3 calls)

C4C = .94702925 (P4's min call hand when P1 calls, P2 calls, P3 folds)

C4D = .96887551 (P4's min call hand when P1 calls, P2 calls, P3 calls)

---

C5A = .89931525 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 folds)

C5B = .94331611 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 calls)

C5C = .94474327 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 folds)

C5D = .96748541 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 calls)

C5E = .94616803 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 folds)

C5F = .96828423 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 calls)

C5G = .96861424 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 folds)

C5H = .98094976 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 calls)

---

C6A = .89569346 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 folds)

C6B = .94085752 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 calls)

C6C = .94233556 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 folds)

C6D = .96601165 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 calls)

C6E = .94381871 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 folds)

C6F = .96684643 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 calls)

C6G = .96720214 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 folds)

C6H = .98008153 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 calls)

C6I = .94529751 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 folds)

C6J = .96767975 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 calls)

C6K = .96801519 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 folds)

C6L = .98056919 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 calls)

C6M = .96835204 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 folds)

C6N = .98076290 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 calls)

C6O = .98086128 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 folds)

C6P = .98811101 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 calls)

---

In my next post I will report the GTO EV's for the players in this game.

Here are the GTO EV's for the players in the respective toy games (without antes) solved so far. The values below are the exact EV's whereas previously I used simulations for a few of the cases. I have now programmed the complete game tree and derived each player's exact EV on each branch of the tree.

Players	P0	P1	P2	P3	P4	P5	P6
2	-0.250000	0.250000
3	-0.399056	0.167931	0.231125
4	-0.496559	0.125292	0.161640	0.209627
5	-0.565278	0.099662	0.122836	0.152215	0.190565
6	-0.616430	0.082656	0.098620	0.118121	0.142637	0.174396
7	-0.656085	0.070577	0.082212	0.096025	0.112792	0.133744	0.160735

Here are three tables which answer your questions. The tables are based upon the results of simulations. Each simulation consisted of 100 million trials using the GTO strategies for each of the toy games from 2 to 7 players.

Players	Pct of Time Overall Best Hand Wins Pot
2	87.50%
3	86.46%
4	86.25%
5	86.11%
6	85.97%
7	85.84%

Players	Pct of the times P0 does NOT Win Pot when he has Overall Best Hand	Pct of the times P1 does NOT Win Pot when he has Overall Best Hand	Pct of the times P2 does NOT Win Pot when he has Overall Best Hand	Pct of the times P3 does NOT Win Pot when he has Overall Best Hand	Pct of the times P4 does NOT Win Pot when he has Overall Best Hand	Pct of the times P5 does NOT Win Pot when he has Overall Best Hand	Pct of the times P6 does NOT Win Pot when he has Overall Best Hand
2	0.00%	25.00%
3	0.00%	25.48%	15.15%
4	0.00%	26.35%	18.12%	10.52%
5	0.00%	27.06%	20.43%	13.55%	8.40%
6	0.00%	27.61%	22.10%	16.23%	10.76%	7.47%
7	0.00%	28.05%	23.37%	18.31%	13.26%	9.07%	7.09%

Players	Pct of the times P0 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P1 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P2 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P3 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P4 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P5 does NOT have Overall Best Hand when he Wins Pot	Pct of the times P6 does NOT have Overall Best Hand when he Wins Pot
2	20.00%	0.00%
3	26.84%	3.20%	1.70%
4	30.56%	4.46%	3.60%	4.76%
5	33.02%	5.09%	4.46%	4.91%	8.10%
6	34.82%	5.45%	4.95%	5.05%	6.72%	11.19%
7	36.20%	5.70%	5.29%	5.21%	6.06%	8.72%	13.89%

I have saved all the raw totals of these simulations so would be able to easily answer any other questions in case I misunderstood what you were asking.

Here is the solution for the 8-player game without antes.

Following the lead from above, label the players P0, P1, P2, P3, P4, P5, P6, and P7 (P0 bets blind 100% of the time). The other players can call or fold. The GTO solution consists of the minimum hand each player calls with on each possible branch of the game tree. As above, the GTO solution for many cases in the 8-player game tree devolve into the lower-order GTO solutions for the smaller games (derived and presented previously in the thread).

The "new" GTO branches are as follows:

---

C1 = .85428329 (P1's min call hand)

---

C2 = .92122710 (P2's min call hand when P1 calls)

---

C3A = .91866516 (P3's min call hand when P1 calls and P2 folds)

C3B = .95467593 (P3's min call hand when P1 calls and P2 calls)

---

C4A = .91603133 (P4's min call hand when P1 calls, P2 folds, P3 folds)

C4B = .95299330 (P4's min call hand when P1 calls, P2 folds, P3 calls)

C4C = .95403058 (P4's min call hand when P1 calls, P2 calls, P3 folds)

C4D = .97298036 (P4's min call hand when P1 calls, P2 calls, P3 calls)

---

C5A = .91333368 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 folds)

C5B = .95123216 (P5's min call hand when P1 calls, P2 folds, P3 folds, P4 calls)

C5C = .95230644 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 folds)

C5D = .97193267 (P5's min call hand when P1 calls, P2 folds, P3 calls, P4 calls)

C5E = .95337839 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 folds)

C5F = .97253406 (P5's min call hand when P1 calls, P2 calls, P3 folds, P4 calls)

C5G = .97278325 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 folds)

C5H = .98347919 (P5's min call hand when P1 calls, P2 calls, P3 calls, P4 calls)

---

C6A = .91058239 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 folds)

C6B = .94939225 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 calls)

C6C = .95050143 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 folds)

C6D = .97082989 (P6's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 calls)

C6E = .95161236 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 folds)

C6F = .97145508 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 calls)

C6G = .97172204 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 folds)

C6H = .98282647 (P6's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 calls)

C6I = .95271983 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 folds)

C6J = .97207901 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 calls)

C6K = .97233166 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 folds)

C6L = .98319243 (P6's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 calls)

C6M = .97258516 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 folds)

C6N = .98333858 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 calls)

C6O = .98341244 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 folds)

C6P = .98969571 (P6's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 calls)

---

C7A = .90778936 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 folds, P6 folds)

C7B = .94747429 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 folds, P6 calls)

C7C = .94861542 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 calls, P6 folds)

C7D = .96967056 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 folds, P5 calls, P6 calls)

C7E = .94976314 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 folds, P6 folds)

C7F = .97031875 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 folds, P6 calls)

C7G = .97060241 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 calls, P6 folds)

C7H = .98213768 (P7's min call hand when P1 calls, P2 folds, P3 folds, P4 calls, P5 calls, P6 calls)

C7I = .95091158 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 folds, P6 folds)

C7J = .97096796 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 folds, P6 calls)

C7K = .97123756 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 calls, P6 folds)

C7L = .98251884 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 folds, P5 calls, P6 calls)

C7M = .97150821 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 folds, P6 folds)

C7N = .98267544 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 folds, P6 calls)

C7O = .98275462 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 calls, P6 folds)

C7P = .98928520 (P7's min call hand when P1 calls, P2 folds, P3 calls, P4 calls, P5 calls, P6 calls)

C7Q = .95205537 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 folds, P6 folds)

C7R = .97161524 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 folds, P6 calls)

C7S = .97187121 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 calls, P6 folds)

C7T = .98289914 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 folds, P5 calls, P6 calls)

C7U = .97212826 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 folds, P6 folds)

C7V = .98304762 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 folds, P6 calls)

C7W = .98312370 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 calls, P6 folds)

C7X = .98951344 (P7's min call hand when P1 calls, P2 calls, P3 folds, P4 calls, P5 calls, P6 calls)

C7Y = .97238614 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 folds, P6 folds)

C7Z = .98319659 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 folds, P6 calls)

C7AA = .98327109 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 calls, P6 folds)

C7AB = .98960364 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 folds, P5 calls, P6 calls)

C7AC = .98334582 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 folds, P6 folds)

C7AD = .98964743 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 folds, P6 calls)

C7AE = .98967206 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 calls, P6 folds)

C7AF = .99346786 (P7's min call hand when P1 calls, P2 calls, P3 calls, P4 calls, P5 calls, P6 calls)

---

In my next post I will report the GTO EV's for the players in this game.

Here are the GTO EV's for the players in the respective toy games (without antes) solved so far. The values below are the exact EV's whereas previously I used simulations for a few of the cases. I have now programmed the complete game tree and derived each player's exact EV on each branch of the tree.

Players	P0	P1	P2	P3	P4	P5	P6	P7
2	-0.250000	0.250000
3	-0.399056	0.167931	0.231125
4	-0.496559	0.125292	0.161640	0.209627
5	-0.565278	0.099662	0.122836	0.152215	0.190565
6	-0.616430	0.082656	0.098620	0.118121	0.142637	0.174396
7	-0.656085	0.070577	0.082212	0.096025	0.112792	0.133744	0.160735
8	-0.687795	0.061564	0.070409	0.080683	0.092819	0.107482	0.125720	0.149118